This paper derives well-posedness and asymptotic results that provide qualitative information about the behavior, mechanism and strategies used by living organisms to navigate their biological networks. Chemical driven swimming is a captivating phenomenon that is observed in various living organisms like bacteria and protozoa but the problem in weighted networks is more complex, since the equations of parabolic-parabolic Keller-Segel model coupled with incompressible Navier-Stokes equations must be reformulated in a discrete setting. The starting point is to transpose the coupled system from the Euclidean case to the connected networks with certain network-theoretic simplified approaches, which yields fruitful key results. These results not only enable the construction of global solutions but also serve as a foundation for determining information about the stability and large time behavior of the system. Then, decay rates are well established to predict important features, such as how quickly weak solutions at a given point decrease over time due to dissipative processes. Additionally, the - convergence of cell densities towards the self-similar Gaussian solution of the heat equation is well proved by time dependent scaling, which shows that the solution maintains its shape and only scales in time and space as time evolves. Finally, this paper includes many numerical tests through a recent robust numerical scheme to illustrate the theoretical results and to develop computational control and prediction of living organisms' trajectories around central nodes in networked flows.

中文翻译：

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加权网络中化学驱动游泳生物体的数学分析和渐近预测


本文得出适定性和渐近结果，提供有关生物体用于导航其生物网络的行为、机制和策略的定性信息。化学驱动的游泳是一种令人着迷的现象，在细菌和原生动物等各种生物体中都可以观察到，但加权网络中的问题更为复杂，因为抛物线-抛物线 Keller-Segel 模型的方程与不可压缩的纳维-斯托克斯方程必须重新表述为离散设置。出发点是通过某些网络理论简化方法将耦合系统从欧几里德情况转置为连通网络，从而产生了富有成效的关键结果。这些结果不仅能够构建全局解决方案，而且可以作为确定有关系统稳定性和长时间行为的信息的基础。然后，衰减率可以很好地预测重要特征，例如由于耗散过程，给定点的弱解随着时间的推移而减少的速度有多快。此外，细胞密度向热方程的自相似高斯解的收敛性通过时间相关缩放得到了很好的证明，这表明该解保持其形状并且仅随着时间的演变而在时间和空间上缩放。最后，本文通过最近的稳健数值方案进行了许多数值测试，以说明理论结果并开发网络流中中心节点周围生物体轨迹的计算控制和预测。